Coalescence of Bubbles and Stability of Foams in Brij Surfactant

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Coalescence of Bubbles and Stability of Foams in Brij Surfactant Systems Sayantan Samanta and Pallab Ghosh* Department of Chemical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, Assam, India ABSTRACT: This work presents coalescence of air bubbles and stability of foams in aqueous solutions of polyoxyethylene (4) lauryl ether (Brij 30), polyoxyethylene (23) lauryl ether (Brij 35), and polyoxyethylene (20) oleyl ether (Brij 98). Adsorption of these surfactants at the air-water interface was also studied. The surface tension profiles were fitted by the Szyskowski equation. It was found that the surface tensions at the critical micelle concentrations of these surfactants followed the sequence Brij 35 > Brij 98 > Brij 30. The air bubbles were most stable to coalescence in Brij 35 solution and least stable in Brij 30 solution. Coalescence in Brij 35 solution was hindered beyond a certain concentration of the surfactant. Stochastic distributions of coalescence time were observed in all surfactant systems. These distributions were fitted well by the stochastic model. Seven film-drainage models were used to predict the coalescence time. However, their predictions did not agree well with the mean values of the coalescence time distributions. Stability of foams was analyzed by the Ross-Miles test. The initial and residual foam heights were measured at each surfactant concentration. The foam heights followed the trends that are expected from the coalescence times of the bubbles.

’ INTRODUCTION Bubbles, in aqueous solutions, have been the subject of numerous studies due to their importance in many industrial, environmental, and biological processes. The presence of gas bubbles is sometimes useful (e.g., heavy metal ions can be efficiently removed from wastewater stream using gas bubbles). However, at times, the presence of bubbles may be undesirable. For example, the presence of gas bubbles affects the quality of finished products in photographic and paper industries. Foaming creates operational problems during acid gas absorption by aqueous alkanolamine solutions.1 The understanding of bubble growth, stability, and coalescence is crucial in many of these processes.2,3 Thus, a sound understanding of coalescence of bubbles in aqueous solutions is essential not only from a fundamental viewpoint but also from an economical perspective. Coalescence of air bubbles is greatly influenced by the type and concentration of surfactants present in the solution. In pure water, air bubbles coalesce almost instantly.4 However, coalescence is not instantaneous in aqueous solutions of surfactants. The main factors that determine the coalescence time in these systems are the surface excess concentration of the surfactant and the repulsive surface forces such as electrostatic double layer, hydration, and steric forces. The role of these forces in the stability of thin liquid films has been discussed in the literature.5,6 Coalescence time of bubbles increases with increase in surfactant concentration. Ionic surfactants stabilize the thin liquid films by electrostatic double layer and hydration forces, whereas the nonionic surfactants stabilize the films by steric force. A review of the early works on coalescence of bubbles and the details of film-drainage theories have been presented by Chaudhari and Hofmann.7 A recent review of the literature on coalescence of bubbles has been presented by Ghosh.8 The nonionic surfactants have diverse applications in pharmaceuticals, cosmetics, and paints. They are electrically neutral. Therefore, they are less sensitive to the presence of electrolytes r 2011 American Chemical Society

(such as mineral salts) in the medium than the ionic surfactants. These surfactants offer a high degree of flexibility for synthesis as well and possess an extremely compatible set of physical properties that allow for widespread use along with other surfactants. The nonionic surfactants are particularly useful where low foaming is required. In recent years, the linear polyoxyethylene (POE) alcohol surfactants (under the commercial name of Brij surfactants) have found widespread use in drug-release systems,9 micellar reactions,10 self-assembled organized structure for biomimicking,11 emulsion polymerization,12 and removal of organic pollutants from water.13 Klammt et al.14 have reported that the Brij surfactants are most suitable for the soluble cell-free expression of membrane proteins. These surfactants contain a hydrophilic head with a varying number of polyoxyethylene groups and a distinct hydrophobic tail consisting of a polymethylene chain. The general structure of these surfactants is CnH2n(1-(O-CH2-CH2)x-OH, where the CnH2nþ1 segment is for the lauryl, cetyl, and stearyl series and the CnH2n-1 segment is for the oleyl series. Many Brij surfactants are stable to acids and alkalis beyond the pH range that the ester-type emulsifiers can withstand. Hence, these are useful for emulsifying fats and oils in highly acidic or alkaline media. As far as we know, there is hardly any work reported in the literature on coalescence of air bubbles in aqueous solutions of the Brij surfactants. With this background, the main objective of the present work was to investigate the coalescence of air bubbles and stability of foams in aqueous solutions of Brij 30, 35, and 98 and correlate these results, because the bubble coalescence time is often considered as an indication of the stability of foam. The compositions of the surfactant solutions were selected from the Received: November 29, 2010 Accepted: February 14, 2011 Revised: February 8, 2011 Published: March 09, 2011 4484

dx.doi.org/10.1021/ie102396v | Ind. Eng. Chem. Res. 2011, 50, 4484–4493

Industrial & Engineering Chemistry Research

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Table 1. Film-Drainage Models of Coalescence Time equation for coalescence timea

based on the model of

μa3:4 ðΔFgÞ0:6 tc, 1 ¼ 1:07 γ1:2 B0:4

Chen et al.17

μa3:4 ðΔFgÞ0:6 tc, 2 ¼ 0:705 γ1:2 B0:4 tc, 3 ¼ 1:046

μa4:5 ΔFg γ1:5 B0:5

Mackay and Mason18

μa4:5 ΔFg tc, 4 ¼ 0:37 1:5 0:5 γ B μa1:75 tc, 5 ¼ 5:202 0:75 0:25 γ B

Hodgson and Woods19

μa4:06 ðΔFgÞ0:84 tc, 6 ¼ 0:79 γ1:38 B0:46

Slattery16

μa4:06 ðΔFgÞ0:84 tc, 7 ¼ 0:44 γ1:38 B0:46 a

B = 1 10-28 J m.

detailed surface tension profiles of their aqueous solutions. These profiles were fitted using a surface equation of state based on Gibbs and Langmuir adsorption equations. The coalescence time distributions were fitted using the stochastic model of Ghosh and Juvekar,15 and the parameters of the model were analyzed with the physical properties of the systems. The mean values of the coalescence time distributions were compared with the predictions of seven film-drainage models. The foam stability was determined by the Ross-Miles test. The initial foam height, residual foam height, and rate of drainage of the foams were analyzed.

’ GENERAL THEORY Adsorption of Brij Surfactant at the Air-Water Interface. For a nonionic surfactant, the variation of surface tension of its solution with its concentration in the solution is given by the Szyskowski equation.

γ ¼ γ0 - RTΓ¥ lnð1 þ KL cÞ

ð1Þ

Equation 1 can be derived from Gibbs and Langmuir adsorption equations.2 It is a simple surface equation of state (EOS) which has two unknown parameters, namely, Γ¥ and KL. These parameters can be obtained by fitting the experimental surface tension versus concentration data. From the value of Γ¥, the minimum surface area per adsorbed molecule can be obtained as Am ¼

1 Γ ¥ NA

ð2Þ

The value of Am depends on the adsorption characteristics of the surfactant at the interface. Since the value of Γ¥ is asymptotic because Γ¥ is obtained by fitting eq 1 to the γ versus c profiles in the region where c is less than the critical micelle concentration (CMC), the value of Am is approximate. However, these values of Am are not far from the true minimum area because the variation of Γ with c becomes small well before the surfactant concentration approaches the CMC. Film-Drainage Theories of Coalescence. The film-drainage models of coalescence of bubbles in surfactant solutions given by Slattery,16 based on the film-drainage models reported in

literature,17-19 are presented in Table 1. The equations in Table 1 were developed for the buoyancy-driven coalescence of a bubble at a flat gas-liquid interface. Several assumptions were made while developing these models, which have been described in detail by Slattery.16 Nikolov and Wasan20 have suggested that thermodynamic stability of the thin liquid film is the main factor that decides the coalescence time in surfactant-stabilized systems. Vrij and Overbeek21 suggested that the thermal fluctuations can corrugate a deformable interface, and in certain cases, the van der Waals force can be strong enough to cause thermodynamic instability in the film. According to them, a film can rupture after its thickness reduces to the critical value. They developed the following equation for the critical film thickness. !1=7 af A H 2 ð3Þ hc ¼ 0:267 6πγΔp The values of hc usually lie between 20 and 60 nm. A stochastic distribution of hc is always observed.22,23 Stochastic Theory of Coalescence. It has been experimentally observed by several workers that the coalescence time of bubbles does not have a single value even under identical experimental conditions (i.e., when the size of the bubbles, temperature of the system, and composition of the solution remain constant), but a wide distribution is omnipresent.8 In order to explain this and to explain some other observations that cannot be explained by the film-drainage theory (e.g., the effects of surfactant and salt on coalescence time), the stochastic model of coalescence was proposed by Ghosh and Juvekar.15 The following equation for the cumulative distribution of coalescence time was developed by them. 2 8 3 0 19 > > > > > > 7 < = C 16 1 B PΓ 1 6 7 B C pffiffiffiB p ffiffi ffi 7 C FðτÞ ¼ 6erf 1 þ erf ¥ 5 @ A> 24 > S S 2 2 > > 2 Γ Γ > > -λ τ i : ; ð1 þ ∑ e Þ i¼1

ð4Þ The dimensionless coalescence time is defined as, τ = t/t, where t is the characteristic diffusion time, which is given by t ¼ Rb 2 =DΓ

ð5Þ

DΓ is the surface diffusivity of the surfactant molecules. Agrawal and Neuman24 have presented the values of DΓ for many surfactants at various states of the monolayer (e.g., gaseous, liquid-expanded, and liquid-condensed states). The radius of the barrier ring, Rb, may be estimated from the equation given by Princen,25 1=2 2 ΔFg Rb ¼ 2a ð6Þ 3γ The dimensionless coalescence threshold, PΓ, is given by Γm a ΔFgγ 1=2 PΓ ¼ ¼ 3 RΓ ðwb fr RÞΓ

ð7Þ

The normalized standard deviation in surface excess, SΓ, is defined as, σΓ SΓ ¼ ð8Þ Γ 4485

dx.doi.org/10.1021/ie102396v |Ind. Eng. Chem. Res. 2011, 50, 4484–4493


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Table 2. Properties of the Surfactants chemical name

formula

HLB

CMC (mol/m3)

γCMC (mN/m)

Brij 35

polyoxyethylene (23) lauryl ether

C12H25(OC2H4)23OH

16.9

0.090

42.0

Brij 98

polyoxyethylene (20) oleyl ether

C18H35(OC2H4)20OH

15.0

0.025

39.4

Brij 30

polyoxyethylene (4) lauryl ether

C12H25(OC2H4)4OH

9.0

0.025

29.3

name of surfactant

It depicts the broadness of coalescence time distribution. Therefore, the stochastic model has two unknown parameters, namely, PΓ and SΓ, which may be obtained by fitting eq 4 to the experimental coalescence time distribution. The variation of PΓ with the physical properties of the system can be explained from eq 7. The fit of the stochastic model to the experimental distributions of coalescence time of bubbles has been good.26,27 Correlations for the Stability of Foams. Several attempts have been made to correlate the initial foam height (H) with the physicochemical properties of the foaming systems. Rosen and Solash28 have found a linear relationship between the initial foam height and 1/γ, based on their data obtained from static foam test. Similar attempts have been made for dynamic foaming systems as well. Pilon et al.29 have developed a correlation ~ average radius of between the steady-state foam height (H), the bubbles in the foam (r) and surface tension, which predicts ~ with capillary, ~ γ/r 2.6. They have also correlated H that H 30 Froude, and Reynolds numbers. Malysa et al. have found that the retention time (defined as the slope of the linear part of the plot of volumetric gas flow rate versus the gas volume contained in the system) varies linearly with the Marangoni elasticity (which is defined as, EM = dγ/d ln A).

’ EXPERIMENTAL SECTION Materials Used. Brij 30 and 98 were purchased from SigmaAldrich (India). Brij 35 was purchased from Merck (India). All these surfactants had 99% purity. They were used as received from their manufacturers. The properties of these surfactants are presented in Table 2. The water used in this study was purified from a Millipore water purification system. Its conductivity was 1 10-5 S/m, and the surface tension was 72.5 mN/m (300 K). Measurement of Surface Tension. Surface tension was measured using a computer-controlled tensiometer [manufacturer: GBX (France), model: ILMS 4, precision: 0.01 mN/m]. The Wilhelmy plate method was used to measure the surface tension. A thin plate made of platinum and iridium was used. It was dipped into the solution, and the vessel containing the liquid was gradually lowered with a very slow speed (∼200 μm/s). The force measured by the balance at the point of detachment was recorded by the computer. The contact angle was zero because water completely wets the plate. Therefore, the surface tension was given by the force per unit wetted perimeter of the plate. The sample vessels and the plate were methodically cleaned before each measurement. The cleaning was done by following the procedure described in the ASTM Standard D1331-89 (2001). The glassware was cleaned by using a fresh chromicsulfuric acid cleaning mixture, followed by a thorough rinsing in distilled water. The platinum ring was rinsed thoroughly in acetone and in distilled water. The ring was allowed to dry and then heated to white in the oxidizing portion of a Bunsen burner flame.

The entire range of surfactant concentration under study was divided into several intervals. This enabled us to detect the subtle changes in surface tension accurately. A stock solution was prepared by dissolving the surfactant in water. Subsequently, this solution was diluted to prepare the surfactant solutions of required concentration. The values of surface tension measured by the procedure mentioned above were highly accurate and reproducible. Measurement of Surface Shear Viscosity. An interfacial rheometer [manufacturer: Anton-Paar (Germany), model: Physica MCR 300] equipped with a biconical bob was used to measure the surface shear viscosity. The temperature was maintained with the help of a Peltier element. The same procedure that was employed for cleaning the vessels during the measurement of surface tension was also used for the measurement of interfacial shear viscosity. Study of Coalescence of Bubbles at Air-Water Interface. Coalescence time of bubbles was studied by following the procedure reported in the literature.31,32 The bubbles were formed in a specially designed coalescence cell made of glass [manufacturer: Schott Duran (Germany)]. The diameter of the cell was 10 cm. The vessel had a hole on its wall near the bottom where a Teflon coated rubber septum was fixed. Air bubbles were formed by a syringe inserted through the septum. The bubbles were released a few centimeters away from the flat air-water interface. The time during which a bubble rested on the flat airwater interface (i.e., the coalescence time) was measured by a digital video camera [manufacturer: Sony (Japan), model: DCRHC32E, optical zoom: 20 ]. It had a timer having a resolution of 0.1 s fitted with it. The time count began as soon as the bubble struck the flat air-water interface. In some cases, especially at the low surfactant concentrations, some bubbles coalesced as soon as they struck the interface, which we call instantaneous coalescence. After a bubble coalesced, the next bubble was released after the visible disturbances at the flat interface had subsided. Coalescence times of 100 bubbles were studied in each experiment. The size of the bubble was determined by image analysis using the ImageJ software.33 The bubble was formed inside the surfactant solution contained in a flat-walled glass vessel (5 cm 5 cm 10 cm) [manufacturer: Remco (India)] following the procedure described before. The photograph of the bubble was taken when it was fully formed at the tip of the needle and was about to be released. The pixel of the image was calibrated with a well-defined length (used as reference). The maximum lengths in the horizontal and vertical directions were measured from the photograph. As the bubble was small, its deformation was small. The diameter of the bubble was taken as the average of these lengths. Ten bubbles were analyzed for each system, and the mean diameter was calculated. Because the surface tension varies with the change in surfactant concentration and the size of the bubble depends on surface tension, the air flow and needle diameter were adjusted to keep the size of the bubble constant. Static Test of Foam Stability. The Ross-Miles foam test was performed by following the procedure described in the ASTM 4486



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Figure 1. Variation of surface tension of aqueous solution of Brij 35 with the concentration of surfactant. The fit of the surface EOS (eq 1) and the variation of Γ with surfactant concentration are shown in the figure.

Standard D1173-07 (2007). The foam was generated by filling a pipet with 200 cm3 of the surfactant solution. The solution was allowed to fall a specified distance (90 cm) into 50 cm3 of the same solution that was contained in a receiver. The height of the foam that was produced immediately upon draining of the pipet was measured by using the digital video camera. The decay in foam height with time was also measured. Experimental Conditions. All experiments were carried out in an air-conditioned room where the temperature was maintained at 300 K. The variation of temperature in the room was within 0.5 K.

’ RESULTS AND DISCUSSION Adsorption of Surfactants at the Air-Water Interface. The adsorption of Brij surfactants at the air-water interface was studied by measuring the variation of surface tension with surfactant concentration. The surface tension profile for Brij 35 is shown in Figure 1. The surface tension became invariant with surfactant concentration when the concentration of surfactant was ∼0.09 mol/m3. There is some difference among the values of CMC of Brij 35 reported in the literature.34,35 The CMC of Brij 35 determined in our study agrees well with the CMC reported by Schwarz et al.34 The surface EOS (eq 1) fitted the surface tension profile well. The two parameters of the surface EOS, namely, Γ¥ and KL, are presented in Table 3. These parameters were obtained by minimization of the mean square average deviation (Δ) in surface tension (based on the pre-CMC data).

∑

Δ ¼ ½ ðγexp - γmodel Þ2 =m0:5

ð9Þ

The values of Δ are reported in Table 3. The surface area occupied by a surfactant molecule at the air-water interface is given in the last column of Table 3. The surface tension attained at CMC, γCMC, is given in Table 2. The surface tension profiles for Brij 98 and Brij 30 are shown in Figures 2 and 3, respectively. The parameters of the EOS, that is, KL and Γ¥, are given in Table 3. The CMCs (Table 2) agree well with those reported by Schwarz et al.34 and Andersson et al.6 The values of γCMC for the three surfactants follow the sequence: Brij 35 > Brij 98 > Brij 30. The values of Γ¥ follow the reverse

sequence. It is evident from these values that there is difference in the composition and structure of the surface phase among these surfactants. The surfactant molecules orient in the monolayer at the air-water interface depending on their structure.36 The hydrocarbon chain (namely, CnH2nþ1) is same for Brij 30 and Brij 35. However, Brij 35 has more epoxy groups, and therefore, its hydrophilic portion is larger. This results in a lower concentration of Brij 35 molecules at the air-water interface (i.e., a lower value of Γ¥) and higher area occupied by a molecule at the interface. Brij 30 has a higher Γ¥ because the area occupied by a molecule at the interface is much smaller. The likely orientations of these surfactants are schematically shown in Figure 4. In addition, Brij 30 shows a much lower γCMC than either Brij 35 or Brij 98. Surface tension of aqueous surfactant solution is mainly determined by the groups and molecules present in the outermost layer of the surface.36 The contribution of water to the surface energy is much higher than the methylene or methyl groups. This indicates that the water content is smaller in the monolayer of Brij 30 than in Brij 35 and 98. A comparison of the CMCs of the three surfactants (Table 2) shows that the CMC of Brij 35 is higher than that of Brij 98 and Brij 30 and that the CMCs of the latter two surfactants are quite similar. The free energy of formation of micelles depends on the hydrophobic and hydrophilic parts of the surfactants. The hydrophobic part of the Brij surfactants is CnH2n(1, and the rest, namely, -(O-CH2-CH2)x-OH, is the hydrophilic part.37 Brij 30 and Brij 35 have the same hydrocarbon chain. The higher CMC of Brij 35 may be attributed to the larger number of epoxy groups present in it, which decreases the ratio of the hydrophobic and hydrophilic contents of the surfactant. In the same homologous series, increase in the length of the hydrocarbon chain usually leads to a reduction in the CMC,38 because formation of micelle becomes easier with increase in hydrophobicity. A comparison of the structures of Brij 35 and Brij 98 (see Table 2) shows that the hydrophobic part is much larger and the hydrophilic part is slightly smaller for the latter. This results in a lower CMC for Brij 98. A comparison of the structures of Brij 98 and Brij 30 shows that both the hydrophobic and the hydrophilic parts of Brij 98 are larger than the corresponding parts of Brij 30. Similar CMCs of these two surfactants indicate that these effects compensate each other toward the net free energy of micelle formation. The surface excess concentration (Γ) was calculated from the Langmuir adsorption equation. KL c ð10Þ Γ ¼ Γ¥ 1 þ KL c The variation of Γ with surfactant concentration (c) is shown in Figures 1-3. The interesting aspect of these profiles is that Γ increases significantly with surfactant concentration when the latter is small, and then the variation becomes very gradual at higher surfactant concentrations. Similar observations can be made from the results reported in the literature.39 The equilibrium constants (KL) presented in Table 3 represent the relative rates of adsorption and desorption of the surfactant molecules. The higher values of KL for Brij 35 and Brij 98 indicate either a fast rate of adsorption of the surfactant molecules at the airwater interface and/or slow rate of desorption of these molecules from the interface. Coalescence of Air Bubbles at the Flat Air-Water Interface. Coalescence of air bubbles was studied at three surfactant 4487



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Table 3. Parameters of the Surface EOS and the Area Occupied by a Surfactant Molecule at the Air-Water Interface (at NearSaturation) KL (m3/mol)

Γ¥ (mol/m2) 106

Δ (N/m) 103

Am (m2) 1020

Brij 35

16400

1.65

0.02

100.6

Brij 98 Brij 30

22954 2158

2.30 5.76

0.60 0.63

72.2 28.8

name of surfactant

Figure 4. Schematic of the orientation of surfactant molecules: (a) high surface coverage and (b) low surface coverage. The space between the head groups of the surfactant molecules is filled by water molecules.

Table 4. Parameters of the Stochastic Model for Coalescence of Air Bubbles at the Flat Air-Water Interface in the Presence of Brij 35, Brij 98, and Brij 30a surfactant Brij 35


Brij 98

Brij 30

c (mol/m3)

t (s)

PΓ

SΓ

0.007

7589.5

16.3

0.12

0.010

7765.7

4.6

0.11

0.015

8040.9

4.2

0.10

0.007

9336.9

18.4

0.14

0.020

10082.5

14.2

0.13

0.022 0.010

10125.8 6292.9

5.3 19.5

0.13 0.05

0.015

6603.4

19.1

0.04

0.025

6860.7

18.6

0.04

Physical properties of the systems: ΔF = 995.9 kg/m3, g = 9.8 m/s2, and a = 1.325 mm. a


concentrations. These concentrations were selected based on the coalescence times of the bubbles. The compositions of the coalescence systems are presented in Table 4. The characteristic diffusion time, t, of the stochastic model was calculated from eq 5, and the radius of the barrier ring, Rb, was calculated from eq 6. The calculation of t requires the value of the surface diffusivity (DΓ) of the surfactant molecules. The value of DΓ, however, is approximate because experimental determination of surface diffusivity involves a considerable amount of complexity. The

values reported in the literature24 often show orders of magnitude difference from one another. Temperature and humidity, which cause extraneous surface convective flows, play important roles in these variations. On the basis of the values reported in the literature, a value of DΓ = 1 10-10 m2/s was used for the surfactants having larger hydrophilic parts (viz., Brij 35 and Brij 98), and DΓ = 2 10-10 m2/s was used for Brij 30, which has a much smaller hydrophilic part. Ghosh and Juvekar15 have pointed out that the stochastic model parameters t and PΓ are autocorrelated. Therefore, a different choice of surface diffusivity would lead to a different value of t and, hence, PΓ. However, in a given set of experiments where the surface diffusivity is constant, the fit of the model and the trend of PΓ are insensitive to the assumed value of DΓ. This implies that although we cannot predict the value of PΓ accurately by this analysis, we can correctly predict its trend. The bubble coalescence time distributions for the Brij 35 system are shown in Figure 5. In dilute surfactant solution (viz., 0.007 mol/m3), the bubbles coalesced within a few seconds. However, as the surfactant concentration was increased, a sudden increase in coalescence time was observed. The threshold concentration was ∼0.015 mol/m3. Beyond this concentration, many bubbles took one minute or more to coalesce and some bubbles did not coalesce at all. The high stability of the bubbles at 4488



Figure 5. Variation of coalescence time with the concentration of Brij 35. Bubble diameter = 2.65 mm.

Figure 6. Schematic representation of the interaction between the adsorbed layers of polymeric surfactant at the bubble surface and flat air-water interface.

this surfactant concentration may be attributed to the steric force imparted by the long-chain hydrophilic part of the surfactant molecules. The polymeric adsorbed layer encounters a reduction in entropy when confined in a very small space as the bubble approaches the flat air-water interface (see Figure 6). Since the reduction in entropy is thermodynamically unfavorable, their approach is inhibited. For the surfactants used in the present study, the steric repulsion originated from the overlap of the adsorbed layers may be expressed by the de Gennes equation.40,41 The repulsive disjoining pressure (Π) between the two surfaces is given by " 3=4 # kT 2L 9=4 δ Π 3 , δ < 2L ð11Þ s δ 2L Equation 11 assumes that L > s. s is related to the surface excess concentration, Γ, by the equation 1 1=2 ð12Þ s¼ ΓNA The first term in eq 11 arises from the osmotic repulsion between the coils, and the second term comes from the elastic energy of the chains. An approximate calculation is presented here to illustrate the effect of steric repulsion. Suppose that the surface is moderately covered such that Γ = 1 10-6 mol/m2, then eq 12 gives s = 1.3 nm. For δ/2L = 0.9, from eq 11, we obtain repulsive pressure, Π = 6.5 105 N/m2. This pressure acts along the

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barrier ring, which has a small width, say, ∼50 nm. Therefore, for a 1 mm radius barrier ring, the area on the barrier ring that provides the major part of the repulsion is 2πRbwb = 3.1 10-10 m2. The magnitude of the repulsive force originated from the overlap of the adsorbed surfactant monolayers is 2πRbwbΠ, that is, 2 10-4 N. This repulsive force is sufficient to balance the buoyancy acting on a 3 mm diameter air bubble (viz., 1.4 10-4 N). Equations 11 and 12 were developed based on the assumption that each of the polymer molecules is grafted at one end to the surface, which may be quite different from the adsorption of the polymeric surfactant molecules at the fluid-fluid interfaces. Nonetheless, the calculations presented here demonstrate that the steric repulsive force exerted by the surfactant molecules is sufficient to balance the buoyancy force by which the bubble is pressed to the flat air-water interface. The fit of the stochastic model (eq 4) to the coalescence time distributions is shown by the lines in Figure 5. The parameters of the stochastic model (i.e., t, PΓ, and SΓ) are presented in Table 4. It is observed from these values that there was a large reduction in the value of the dimensionless coalescence threshold, PΓ, when the surfactant concentration was √ increased from 0.007 to 0.015 mol/m3. From eq 7, PΓ ( γ)/(frΓ h), because the other parameters in this equation do not vary significantly with surfactant concentration. The variation of surface tension in this concentration range was small. The variation in the surface excess concentration was also very small, as observed from Figure 1. Therefore, the main factor that causes reduction in the value of PΓ is the repulsive force. As a result of the steric force exerted by the surfactant molecules (schematically illustrated in Figure 6), the bubbles are prevented from coalescence. It is noted that the high coalescence time was observed at surfactant concentrations well below the CMC of the surfactant due to this repulsion. A similar abrupt increase in coalescence time was observed for the Brij 98 system as well (Figure 7). In this system, however, the transition concentration was ∼0.022 mol/m3, which is close to the CMC of the surfactant. At 0.007 and 0.02 mol/m3 concentrations of the surfactant, the bubbles coalesced within a few seconds. The values of PΓ shown in Table 4 indicate a decrease with increase in surfactant concentration similar to that observed in the Brij 35 system. The coalescence time distributions in aqueous solutions of Brij 30 are shown in Figure 8. The coalescence times were small for this surfactant at all concentrations. The coalescence studies up to the CMC showed no appreciable stabilization of the bubbles by the surfactant. It is likely that the steric repulsive force that stabilizes the bubbles in 4489




presence of Brij 35 and Brij 98 is absent for Brij 30, because the hydrophilic part of Brij 30 is much smaller than that of Brij 35 and Brij 98. Therefore, although the adsorption of Brij 30 at airwater interface leads to low values of γCMC and its surface excess concentration is high, it is not effective in stabilizing the bubbles. Evidently, the parameters such as surface tension and surface excess concentration cannot completely predict the magnitude of coalescence time. It is noted from Table 4 that the values of PΓ do not change significantly with the concentration of Brij 30. The values of the normalized standard deviation, SΓ, were small for all three surfactant systems, especially Brij 30. The small value of SΓ reflects that the ratio of the standard deviation (σΓ) to the mean of the Gaussian distribution of surface excess concentration (Γh) was small. In other words, the fluctuations in Γ from one bubble to another were small in these surfactant systems. Furthermore, the values of SΓ did not vary significantly with surfactant concentration, which reflects that the surface inhomogeneity did not change with surfactant concentration. The small fluctuation reflects the stability of the monolayer, which is probably due to extensive hydration of the hydrophilic part of the surfactant molecules. The coalescence times predicted by seven film-drainage models are presented in Table 5. The mean values of the coalescence time distributions are given in the last column of this table. According to these models, coalescence time is correlated to surface tension as tc γ-p, where p varies between 0.75 and 1.5. The values predicted by all the models are quite different from the experimental value. None of these filmdrainage models take into account the effects associated with the steric force. It is evident from the present work that the steric force provides a significant contribution to the stability of the bubbles against coalescence for Brij 35 and Brij 98. According to the hydrophobic-force theory of Wang and Yoon,42 the air bubble is most hydrophobic in absence of surfactant, and its hydrophobicity decreases with increase in surfactant concentration. The decrease in hydrophobic force is related to the adsorption of surfactant at the air-water interface. The hydrophobic force becomes weaker with increase in the surface excess concentration. The pure air-water interface is most hydrophobic, and coalescence of air bubbles is instantaneous in pure water. From the results obtained in this study, it is apparent that the coalescence time data for a given surfactant qualitatively obey this theory. The surface excess concentrations of the three surfactants follow the sequence: Brij 30 > Brij 98 > Brij 35. However, the

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stability of bubbles follows the reverse sequence as shown in Figures 5, 7, and 8. Therefore, it is evident that the structural features of the surfactants are important in the coalescence time of bubbles. Stability of Foams in Aqueous Solutions of Brij 35, Brij 98, and Brij 30. The variations of foam height with time at different surfactant concentrations are depicted in Figures 9-11 for the three surfactant systems. Several works have attributed the initial foam height as a parameter of foam formation,28,43 because the foam generated by the Ross-Miles technique is a dynamic phenomenon involving rapid entrainment of air. The other parameter studied in the Ross-Miles test is the residual foam height. It is observed from Figures 9-11 that the initial foam height increased with increase in surfactant concentration. The initial foam height followed the sequence Brij 35 > Brij 98 > Brij 30. These results agree with the mean values of bubble coalescence time distributions presented in Table 5. The foam height decreased rapidly with time during the initial period (viz., the first 100 seconds), and the residual foam height changed slowly with time thereafter. The rapid drainage can be attributed to the dynamic adsorption of surfactant molecules at air-water interface. It has been reported in the literature44 that the rate of adsorption of a nonionic surfactant is related to its CMC: a surfactant with higher CMC undergoes faster adsorption at the interface. Therefore, Brij 35, which has higher CMC than Brij 98 and Brij 30, produces a higher initial foam height. However, the initial foam height for this surfactant did not increase beyond 100 mm even when its concentration was increased to the CMC, but the residual foam height increased considerably. This clearly indicates that the stability of the residual foam increased with increase in surfactant concentration even though the efficiency in foam formation (as measured by the initial foam height) did not increase beyond 0.015 mol/m3 concentration of Brij 35. The initial and residual foam heights in Brij 98 systems were significantly lower in comparison with Brij 35. The foam heights in the Brij 30 system were much smaller than the Brij 98 system. The foam heights increased with increase in surfactant concentration up to the CMC for both of these surfactant systems. The residual foam heights in all these surfactant systems follow similar trends as the coalescence times of bubbles shown in Figures 5, 7, and 8. As per the film-drainage theory, the stability of foam depends on the Marangoni effect, surface shear viscosity, and surface elasticity. Some film-drainage theories suggest that the effect of surface viscosity is negligible as compared to the Marangoni effect.45 The relation between foam stability and surface viscosity is not clear because there are stable foams in which the surface viscosity is not particularly high, and there are viscous monolayers which do not produce stable foams. The foam is unstable if the surface viscosity is either too high or too low.38 The variation of surface shear viscosity with surfactant concentration is shown in Figure 12. The surface viscosity increased with increase in surfactant concentration and the number of epoxy groups present in the surfactant molecule. Therefore, the stability of foams qualitatively agrees with the results shown in Figure 12. The film elasticity decreases with increase in the number of epoxy groups in the surfactant molecule.44 The formation of foam is favored when the surface elasticity is high because surfactant molecules are to be transported rapidly to the expanding surface. Therefore, the initial foam height would be high if the surface elasticity is high. However, during foam-keeping (which is 4490



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Table 5. Comparison of the Predictions of Coalescence Time by the Film-Drainage Models with the Mean Values of Coalescence Time Distributionsa surfactant Brij 35

Brij 98

Brij 30

a

c (mol/m3)

γ (mN/m)

tc,1 (s)

tc,2 (s)

tc,3 (s)

tc,4 (s)

tc,5 (s)

tc,6 (s)

tc,7 (s)

0.007

52.9

234.6

154.6

9413.7

3329.9

4.3

1609.1

896.2

2.2

0.010 0.015

51.7 49.9

241.2 251.5

158.9 165.7

9743.3 10266.0

3446.5 3631.3

4.4 4.5

1660.8 1742.6

925.1 970.6

65.4 86.8

0.007

43.7

300.8

198.2

12845.2

4543.7

5.1

2141.7

1192.8

2.6

0.020

39.8

329.9

217.4

14414.2

5098.7

5.4

2381.3

1326.2

4.9

0.022

39.7

331.6

218.5

14507.1

5131.5

5.4

2395.4

1334.1

75.5

0.010

31.9

430.5

283.6

20102.9

7110.9

6.3

3233.8

1801.1

1.4

0.015

30.4

456.1

300.5

21608.9

7643.7

6.5

3456.1

1924.9

1.5

0.025

29.3

477.5

314.6

22884.1

8094.7

6.7

3643.3

2029.1

1.6

texpt c (s)

Physical properties of the systems: ΔF = 995.9 kg/m3, μ = 1 10-3 Pa s, g = 9.8 m/s2, B = 1 10-28 J m, and a = 1.325 mm.

Figure 11. Variation of foam height with time in Brij 30 solutions. Figure 9. Variation of foam height with time in Brij 35 solutions.

Figure 12. Variation of surface shear viscosity with the concentration of Brij surfactants.

Figure 10. Variation of foam height with time in Brij 98 solutions.

measured by the residual foam height), the surface should be more rigid, so that film-drainage is slowed down. Therefore, greater stability of residual foams of Brij 35 and Brij 98 as compared to Brij 30 is expected. However, by the same analysis, the initial foam height for Brij 30 is expected to be higher than that of Brij 35, which is opposite to that observed in Figures 9 and 11. The Marangoni effect is significant when the maximum rate of ~ is the dynamic decrease in surface tension, (dγ ~/dt)max (where γ surface tension) is large. The value of (dγ~/dt)max depends on the structure of surfactant and its concentration. For a constant hydrophobic part, it increases with increase in the number of the

~/ epoxy groups present in the surfactant.44 The magnitude of (dγ dt)max increases with increase in surfactant concentration as well.46 Brij 35 has more epoxy groups than Brij 30, which leads to a higher value of (dγ~/dt)max for the former. To summarize, the film-drainage theory predicts that higher magnitudes of interfacial shear viscosity and greater Marangoni effect would stabilize the Brij 35 foams more than the Brij 30 foams, and the foam stability would increase with increase in surfactant concentration for a particular surfactant. These predictions qualitatively agree with the results shown in Figures 9-11.

’ CONCLUSIONS On the basis of the experimental results and the analysis of the theoretical models, we can draw the following conclusions. (1) The CMC of Brij 35 is greater than that of Brij 98 and Brij 30, and the latter surfactants have similar CMCs. These 4491



(2)

(3)

(4)

(5)

are attributed to the structural features of these surfactants such as the number of oxyethylene groups and length of the hydrocarbon chain present in the surfactant molecules. The surface tension at CMC follows the sequence Brij 35 > Brij 98 > Brij 30. This difference is due to the adsorption patterns of these surfactants at the air-water interface, which are also related to the structure of these surfactants. The surface excess concentrations of these surfactants follow the reverse trend. The surface tension profiles are fitted well by the surface EOS developed from the Gibbs and Langmuir adsorption equations. The parameters of the surface EOS are consistent with the adsorption patterns of the surfactants. The area occupied by a Brij 35 molecule at the air-water interface, calculated from the EOS parameter, is largest, and that for Brij 30 is the lowest. The air bubbles are most stable to coalescence in Brij 35 solution and least stable in Brij 30 solution. The coalescence time of air bubbles increases with increase in surfactant concentration. Very high coalescence times are observed at moderate concentrations of Brij 35 (well below its CMC). This high stability is attributed to the disjoining pressure due to steric force. Stochastic distributions of coalescence time are observed in all surfactant systems. The stochastic model of coalescence fits these distributions well, and the parameters of the model are consistent with the properties of the system. The predictions of film-drainage models do not compare well with the mean values of coalescence time distributions. It is observed that a low value of surface tension does not ensure high coalescence time, as predicted by the film-drainage models. A high surface excess concentration also does not lead to high coalescence time. The initial and residual foam heights obtained by the Ross-Miles foam test agree well with the trends observed in the coalescence of bubbles. The predictions from the film-drainage theories qualitatively agree with the stability of foams. The increase in interfacial shear viscosity and (dγ ~/dt)max with surfactant concentration and their variation with the structure of the surfactants seem to be the important factors in the stability of foams.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Tel: þ91 361 2582253. Fax: þ91 361 2690762.

’ ACKNOWLEDGMENT The authors thank the Department of Science and Technology (Government of India) for partial financial support of the work reported in this article. ’ NOMENCLATURE a = radius of bubble (m) af = area of the film (m2) A = surface area (m2) AH = Hamaker’s constant (J m) Am = minimum surface area occupied by a surfactant molecule (m2)

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B = modified Hamaker’s constant (J m) c = concentration of surfactant in the solution (mol/m3) DΓ = surface diffusivity of the surfactant molecules (m2/s) EM = Marangoni elasticity (N/m) fr = repulsive force generated by 1 mol of surfactant at the barrier ring (N/mol) F(τ) = cumulative probability distribution of coalescence time g = acceleration due to gravity (m/s2) hc = critical film thickness (m) H = initial foam height (m) ~ = steady state foam height (m) H k = Boltzmann’s constant (J/K) KL = equilibrium constant for adsorption and desorption of surfactant (m3/mol) L = thickness of polymer brush layer (m) m = number of data points n = number of carbon atoms in the hydrocarbon chain of surfactant NA = Avogadro’s number (mol-1) PΓ = dimensionless coalescence threshold r = average radius of the bubbles in foam (m) R = gas constant (J mol-1 K-1) Rb = radius of barrier ring (m) s = mean distance between the attachment points (m) SΓ = normalized standard deviation t = time (s) t = characteristic diffusion time (s) tc,i = coalescence time predicted by model i (s) T = temperature (K) wb = width of the barrier ring (m) x = number of epoxy groups in the surfactant Greek Letters

r = fraction of the total amount of surfactant at the air-water interface that remains at the barrier ring after the displacement of surfactant molecules to the barrier ring γ = surface tension of surfactant solution (N/m) γ~ = dynamic surface tension of surfactant solution (N/m) γ0 = surface tension of pure water (N/m) γCMC = surface tension at the CMC (N/m) γexp = experimental value of surface tension (N/m) γmodel = value of surface tension predicted by the surface EOS (N/m) Γ = surface excess concentration of the surfactant (mol/m2) Γ¥ = adsorption capacity of the surfactant (mol/m2) Γh = mean value of the distribution of surface excess, Γ (mol/m2) Γm = minimum value of the surfactant concentration at the barrier ring required to prevent the bubble from coalescence (mol/m2) δ = separation between two surfaces (m) Δ = mean square average deviation (N/m) Δp = excess pressure in the film (Pa) ΔF = difference in density between the two phases (kg/m3) λi = roots of the Bessel function of first kind and order one μ = viscosity of the aqueous phase (Pa s) Π = repulsive disjoining pressure due to steric force (Pa) σΓ = standard deviation in the distribution of Γ (mol/m2) τ = dimensionless coalescence time Abbreviations

CMC = critical micelle concentration (mol/m3) HLB = hydrophilic-lipophilic balance 4492



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Coalescence of Bubbles and Stability of Foams in Brij Surfactant

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